Theorems · Theorem · measure theory

essSup_comp_le_essSup_map_measure

∀ {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α}
  [inst : ConditionallyCompleteLattice β] {γ : Type u_3} {mγ : MeasurableSpace γ} {f : α → γ} {g : γ → β},
  AEMeasurable f μ →
    autoParam (Filter.IsCoboundedUnder (fun x1 x2 => x1 ≤ x2) (MeasureTheory.ae μ) (g ∘ f))
        essSup_comp_le_essSup_map_measure._auto_1 →
      autoParam (Filter.IsBoundedUnder (fun x1 x2 => x1 ≤ x2) (MeasureTheory.ae (MeasureTheory.Measure.map f μ)) g)
          essSup_comp_le_essSup_map_measure._auto_3 →
        essSup (g ∘ f) μ ≤ essSup g (MeasureTheory.Measure.map f μ)

Defined in

Mathlib.MeasureTheory.Function.EssSup

Cited by

2 results in Mathlib

Foundations

Depth 202 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

ConditionallyCompleteLattice